Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 26

This is known as a WT of x ( t ) at scale s and position u , whereas the fac-
tor 1 / √ s is a normalization factor to preserve the energy in the transform.
The quantity Wx ( u, s ) is sometimes known as the WT coecient. A large
correlation or coecient value at some scale s shows that the signal x ( t )is
closely related to the wavelet function. Conversely, small coecients indicate
a lower correlation of the signal with the wavelet function. Using the position
u , we can also determine the position of the strongest correlation in the sig-
nal, then using the WT coecients to determine the type of frequencies in the
signal and where in the signal train they occur, thereby giving us localization
information of the signal x ( t ) in time and frequency. The transform (Equation
2.30) is known as the continuous wavelet transform (CWT) because the signal
x ( t ) is assumed to be continuous on the time domain t .
A wavelet function is usually chosen based on whether it satisfies several
properties, some of which are listed in the following:

a. The wavelet ψ is a function of zero average (Mallat, 1999):

∞

ψ ( t )d t = 0 (2.31)

−∞

b. Property (a) can be generalized to the following zero moments

property:

∞

t n ψ ( t )d t = 0

(2.32)

−∞

for n =0 , 1 ,...,N 1. This property describes the polynomial degree

of the wavelet function, which depicts the relationship of the wavelet
to the slope of the bandpass filter characteristics in the frequency
domain.

−

c. The wavelet function can be smooth or discontinuous. Smoothness is

indicated by the regularity of the function where it is p regular if it is
p times differentiable.

d. The wavelet function is symmetrical about u if

ψ ( t + u )= ψ ( − t + u ) (2.33)

This property is required if the wavelet coecients are to be used to
reconstruct the original signal.

In the case of discrete signals x ( n ), the discrete wavelet transform (DWT) is
used. A dyadic grid for the scale s and translation u parameters is frequently
employed and is defined as follows:

( s j ,u k )=(2 j ,k 2 j : j, k

∈ Z ) (2.34)